Sharp Remez type inequalities estimating the $L_q$ -norm of a function via its $L_p$ -norm

Abstract

UDC 517.5 For any $q\geq p>0,$ $\alpha=(r+1/q)/(r+1/p),$ $f_p\in[0,\infty],$ $\beta\in[0,2\pi),$ we prove the sharp Remez type inequality $$\|x\|_q\leq\frac{\|\varphi_r+c\|_q}{\|\varphi_r+ c\|^{\alpha}_{L_p([0,2\pi]\setminus B_{y(\beta)})}}\|x\|^{\alpha}_{L_p([0,2\pi]\setminus B)}\|x^{(r)}\|^{1-\alpha}_\infty$$ for $2\pi$-periodic functions $x\in L_\infty^r$ that have zeros and satisfy the condition $$\begin{gather}\|x_+\|_p\,\|x_-\|^{-1}_p=f_p,\quad (1)\end{gather}$$ where $\varphi_r$ is Euler's perfect spline of order $r;$ the number $c$ is chosen in such a way that the function $x=\varphi_r+c$ satisfies the condition (1); $B$ is an arbitrary measurable set such that $\mu B\leq\beta\left(\|\varphi_r+c\|_p\left\|x^{(r)}\right\|_\infty\|x\|^{-1}_p\right)^{-1/(r+1/p)},$ the set $B_{y(\beta)}$ is defined by $B_{y(\beta)}:=\{t\in[0,2\pi]\colon|\varphi_r(t)+c|>y(\beta)\},$ and moreover, $\mu B_{y(\beta)}=\beta.$

We also establish sharp Remez type inequalities of various metrics for trigonometric polynomials and for polynomial splines satisfying (1).